Fine boundary regularity for fully nonlinear mixed local-nonlocal problems

TL;DR

This paper establishes boundary regularity results for solutions to fully nonlinear mixed local-nonlocal PDEs, including Lipschitz continuity and boundary H"older regularity, using sub/supersolutions and Harnack inequalities.

Contribution

It provides the first global Lipschitz and boundary regularity results for fully nonlinear mixed local-nonlocal operators in non-translation invariant settings.

Findings

Global Lipschitz regularity of solutions

Boundary H"older regularity of the gradient

Application to boundary regularity of derivatives

Abstract

We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset \mathbb{R}^d,$ let $u\in C(\mathbb{R}^d)$ be a viscosity solution of such Dirichlet problem. We obtain global Lipschitz regularity and fine boundary regularity for $u$ by constructing appropriate sub and supersolutions coupled with a Harnack type inequality. We apply these results to obtain H\"{o}lder regularity of $Du$ up to the boundary.